Project-Mathematics of Crime-fighting

Hello everyone! I'm a first-year math major, just bored out of my mind. So I thought of doing a project in math, to ease my boredom, and learn something new. My level of math would be Calc I, II, and basic ODEs.

Well, what I had in mind was to investigate if math can be used to fight crime (and it amazingly can!). I got this idea from the fact that the police of the city of Santa Cruz have successfully employed a mathematical model against burglaries, and reduced it by around 30% or so.

But I reckon the level of math employed in the above paper is unscalable at my present level (I couldn't get a single word in the paper, though the equations looked beautiful!). So what I was wondering was whether an alternative model could be built, perhaps at a level of math that is definitely higher than what I'm aware of, but yet reachable if I put effort? I'm aware that statistical models can be built, i.e, assigning probability distributions based on crime data, but that's not exactly what I had in mind.

So can anyone give me ideas on how an alternative model can be built? Or if you have any other interesting problems requiring new & challenging mathematical thought, that would do well as a project too!

Nice idea for a project. A couple years ago, I saw a speaker from UCLA (Andrea Bertozzi) give a conference talk on some mathematics she was doing and, if I remember right, the idea was to help the LA police department distribute its beat cops to the right places at the right times. Since you asked I found a link to the project home page.

I don't remember what kind of math it uses, but if you are interested in things like this, then a good foundation in statistics is probably important even if it is not exactly how you want to approach the problem.

Anyway, this sounds like a big open ended project for you. It is probably something that you should talk about with an advisor in your math or applied math department. If you can come up with a decent project, you might be able to do it as an independent study and get credit.

I'm aware that statistical models can be built, i.e, assigning probability distributions based on crime data, but that's not exactly what I had in mind.

You should clarify what you do have in mind - or at least specify things you definitely don't. For example, computer simulations can be used to model complicated phenomena, but to have an interest in such simulations you would need to be interested in programming or in learning particular higher level software that generates such simulations.

If your idea is to do something practical just by reading about a new mathematical topic and doing pencil and paper calculations, this daydream is hard to fulfill. Lots of people in academia are looking for such problems and the ones that are easy to find are quickly found, worked, and published.

You haven't revealed whether you are a keen experimenter. Are you interesting in observing and measuring everyday pheonomena? Or do you find observation and measurement to be tedious and boring?

Thank you, Vargo for the reply! I checked out the link you mentioned, and it has given me some info concerning the various "crime attraction" factors to be weighed in, while considering a mathematical model. I'll have a talk with my math faculty about this, *but I haven't decided on this being the project, the concern mainly being that at my level of math, the model's probably gonna be a "statistically-oriented" one. I'm sorry, but Statistics and me don't get along very well; the more I delve into it, the more it seems like just rote memorisation of formulae & bookkeeping. I'd much rather enjoy making a model based on an analytical, "algorithmic" approach. Thank you for all the info!

Thank you, Stephen Tashi for your reply as well! I realize that I was quite vague in my first post, and I apologise for it. I'll first answer your questions, and add where necessary.

A vital point I hadn't mentioned (thanks for pointing it out) is that I do enjoy programming, and am quite versed in GUI-based Java. I've created programs that verify values, for unsolved number theory problems, like the Collatz conjecture, Lychrel numbers, etc. But I haven't had any experience with simulation-based programming. I'd love to try that out, if you could suggest me a starting point.

My idea of a project is working on ANY problem that challenges me to think up something new mathematically, anything that prompts me to look up a new topic in math, and then come back to the problem and see how I can apply it there, I.e essentially a back-and-forth process between learning something new and applying it. The problem does not have to be practical; actually I've worked so far only in pure math. I just happened to get interested in this crime-modelling problem. Thought it'd be nice to try something I haven't done before.

I'm not doing projects for the sake of getting them published. I do it for the intellectual excitement it gives me. Math problems get me high, haha.

Well, so far, I've been a "pen-and-paper" guy, but as I said, I'm willing to try something new, as long as it challenges me.

As for experimentation, I'm more of a "mathematical experimenter" than a "physical" one; I enjoy experimenting on paper, like checking out a conjecture, trying to find patterns, observing where and how counterexamples can pop up, building results on that, etc. I haven't ever been interested in physical experimentation and measurement. Frankly, I do consider them tedious and boring. I'd rely on readily-available data from experiments done by others, and try to formulate patterns and results from that.

I hope that makes things much more clear, Stephen Tashi. Thank you again, for your time! * *

I am glad to have helped. However, I would like to dispell one notion that you wrote down. You said that statistics seems like rote memorization or somesuch nonsense. I am not by any means an expert in statistics, but if you look around this site and read some of the posts (like Stephen Tashi's, for example), then you'll know that statistics is much much more than that.

All the same, stats are not for everybody, so if it isn't your cup of tea, no worries. But give it a chance :). Besides, if you are going into a STEM field of some kind, you should know it on a basic level anyway.

Pardon this generalizing digression, but since you have an abstract mind, it may interest you.

In his Phd. Thesis (available on the web as "Notes on the Synthesis Of Form" or a similar title) the architect Christopher Alexander discussed the generalities of the problem of designing things.

My interpretation of what he said is:

Some things are designed by beginning work on them and changing their features as the need arises. An example is a primitive shelter built of mud and sticks in a civilzation without zoning codes. The owner can easily tear down walls, expand rooms etc.

Some things must be (more or less) completely designed before they are built and cannot be easily modifed once constructed. This is a difficult problem. Designers face multiple objectives. Some objectives are apparently contradictory. For example, a computer desk needs to accomodate a keyboard at a low enough height so the user doesn't develop carpal tunnel syndrome, but it needs to have enough space under the desk so the user doesn't bump his knees on things. It is important for designers to identify apparently contradictory goals and find a design that either reaches a compromise or shows in a clever way that the two goals are not really contradictory.

Your problem is to design a project. I'd classify it as something that can be easily modified. Nevetheless its interesting to think about it as if the design had to be done in advance.

My impression of your goals
1) It should involve mathematics that is new, at least to you.
2) It should not involve tedious data collection and clerical work
3) It should have some application to crime fighting

I think 2) and 3) may be contradictory goals, depending on what you mean by "crime fighting".

Actual crime fighting would, in some way, actually fight crime. I think any project that was actually applied to fight crime would have to involve dealing with actual data of various sorts. That would probably involve clerical work, hunting for various databases, understanding their formats.

You might prefer some kind of make-believe crime fighting where you make up your data. In my experience, models created without considering what data is actually available have no practical use. However, people who create such models claim they give some qualitative insights.

Suppose we add another goal:
4.The project should use computer simulation(s)

I don't see this goal as contradictory to any of the others, since you already have a tolerance for any clerical work associated with computer programming. Simulations can have various "resolutions". For example, you could have a high resolution simulation of a neighborhood where individual people are represented and their actions (such as going to work, sleeping, eating) are represented in space and time. Such detailed simulations tend to be a big pile of conjectures so you would be making up most of the data. Usually such simulations are "event driven". For example, there would be some probability distribution for how long a person sleeps. When one event (such as "go to sleep") occurs, a random draw is used to schedule other events ('such as "wake up"). The events are kept in a que and the simulation works by processing the event scheduled for the nearest future time.

If you want to use differential equations the most natural simulations would be low resolution and deterministic. For example, you might represent society and neighborhoods in some aggregated manner.

There is a field of mathematics called "stochastic differential equations" that combines deterministic and probabilistic aspects. It's often seen applied to financial modeling. Howevr, it would not be an easy field to jump into. Perhaps you also have the goal

I am glad to have helped. However, I would like to dispell one notion that you wrote down. You said that statistics seems like rote memorization or somesuch nonsense. I am not by any means an expert in statistics, but if you look around this site and read some of the posts (like Stephen Tashi's, for example), then you'll know that statistics is much much more than that.

All the same, stats are not for everybody, so if it isn't your cup of tea, no worries. But give it a chance :). Besides, if you are going into a STEM field of some kind, you should know it on a basic level anyway.

Good luck in your studies.

Ah, I do realise that Statistics is quite indispensable, and can even be fun at times (I especially love those counterintuitive results that can only be made sense with Statistics!). But it's just that the way I've been exposed to it so far, it has just been a bunch of formulae and "plug-and-chug". I guess I should've gone a bit further with it myself though. I'll definitely give it a try when I'm in the mood. Thank you!

Your problem is to design a project. I'd classify it as something that can be easily modified. Nevetheless its interesting to think about it as if the design had to be done in advance.

My impression of your goals
1) It should involve mathematics that is new, at least to you.
2) It should not involve tedious data collection and clerical work
3) It should have some application to crime fighting

I think 2) and 3) may be contradictory goals, depending on what you mean by "crime fighting".

Actual crime fighting would, in some way, actually fight crime. I think any project that was actually applied to fight crime would have to involve dealing with actual data of various sorts. That would probably involve clerical work, hunting for various databases, understanding their formats.

You might prefer some kind of make-believe crime fighting where you make up your data. In my experience, models created without considering what data is actually available have no practical use. However, people who create such models claim they give some qualitative insights.

Suppose we add another goal:
4.The project should use computer simulation(s)

I don't see this goal as contradictory to any of the others, since you already have a tolerance for any clerical work associated with computer programming. Simulations can have various "resolutions". For example, you could have a high resolution simulation of a neighborhood where individual people are represented and their actions (such as going to work, sleeping, eating) are represented in space and time. Such detailed simulations tend to be a big pile of conjectures so you would be making up most of the data. Usually such simulations are "event driven". For example, there would be some probability distribution for how long a person sleeps. When one event (such as "go to sleep") occurs, a random draw is used to schedule other events ('such as "wake up"). The events are kept in a que and the simulation works by processing the event scheduled for the nearest future time.

If you want to use differential equations the most natural simulations would be low resolution and deterministic. For example, you might represent society and neighborhoods in some aggregated manner.

There is a field of mathematics called "stochastic differential equations" that combines deterministic and probabilistic aspects. It's often seen applied to financial modeling. Howevr, it would not be an easy field to jump into. Perhaps you also have the goal

5. The new mathematics should learn-able at a comfortable rate.

I like your way of looking at this, Stephen Tashi. You're quite accurate, though I'm not considering (3) as a goal; Crime-fighting is just one idea that popped in my mind. I'd ready to tackle ANYTHING that is new AND challenging (I emphasize "challenging", because learning new math alone, need NOT necessarily be challenging. So, this can be made as (3) ).

Yes, I'd agree with (4) and (5) as well. And oh, from your post, I see how probabilistic aspects are quite inevitable for any high-resolution simulation (Perhaps it's about time I stepped my foot into the water, and learn the essentials of Stats, the right way). But yes, I'd like to combine deterministic aspects along with it too. I looked a bit about stochastic differential equations, but taking (5) into consideration, I doubt it's doable.

Well, taking all goals into consideration, what would be your optimal suggestion, Stephen Tashi? (I'm sorry, but I'm still blank in terms of ideas!)

If treating real-life situations (or quasi real-life situations) isn't a goal, I suggest picking a topic in numerical methods. Two main divisions of numerical methods are 1) Numerical methods for problems in a continuum, such as solving differential equations, doing definite integrals, etc. 2) Numerical methods for linear algebra, such as finding eigenvalues of matrices.

That sort of project fits well with learning new mathematics and with programming. It doesn't necessarily connect the numerical problem being solved with any practical situation.

The recent thread https://www.physicsforums.com/showthread.php?t=656272 [Broken] (which I couldn't answer) prompted me to look up Richardson Extrapolation. I was surprised that many particular numerical techniques are merely special cases of Richardson Extrapolation. As a specific suggestion, try implementing Richardson Extrapolation. Implementing it as an abstract Java class might not be the efficient way to perform it, but you'd get a good understanding of how the general Richardson_Extrapolation class gives rise to numerical methods known by other names.

If you do something involving probability, I advise studying probability, not introductory frequentist statistics. (Being a probabilist or a stochastic modeler is very different from being a statistician.) I agree that introductory statistics tends to be a pile of obfuscating terminology and rote procedures. (Statistics is a subjective subject. I have a Bayesian outlook.)